A Four-Quadrant Thrust Estimation Scheme for Marine ... - NTNU
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A Four-Quadrant Thrust Estimation Scheme for Marine Propellers: Theory and Experiments Luca Pivano, Tor Arne Johansen and Øyvind N. Smogeli
Abstract— A thrust estimation scheme for marine propellers that can operate in the full four-quadrant range of the propeller shaft speed and the vessel speed has been developed. The scheme is formed by a nonlinear observer to estimate the propeller torque and the propeller shaft speed and by a mapping to compute the thrust from the observer estimates. The mapping includes the estimation of the propeller advance ratio. The advance speed is assumed to be unknown, and only measurements of shaft speed and motor torque have been used. The robustness of the scheme is demonstrated by Lyapunov theory. The proposed method is experimentally tested on an electrically driven fixed pitch propeller in open-water conditions, in waves and with a wake screen that scales the local flow down in order to simulate one of the effects of the interaction between the propeller and the vessel hull. Index Terms— estimation, nonlinear, marine propulsion.
I. I NTRODUCTION In marine guidance, navigation and control (GNC) systems, the low level thruster controllers have traditionally received less attention compared to the guidance system and the highlevel plant control. In the design of Dynamic Positioning (DP), thruster assisted Position Mooring (PM) and autopilot systems, for example, much effort has been put into the high-level control schemes and the propeller dynamics has often been neglected. More recently, also the issue of thruster dynamics and control has received more attention. For recent references, see for example [1], [4], [8], [12], [20], [27], [29], [31], [32] and the references therein. The main difficulties in the design of effective propeller controllers lie in the modeling of the propeller’s dynamics and in the problem of measuring the environmental state. When a ship performs a marine operation, propellers are often affected by thrust losses due to cross flow, ventilation, in-and-out-of water effects, wave-induced water velocities, interaction between the vessel hull and the propeller and between propellers. Propellers may thus work far from ideal conditions therefore, knowledge of the propeller thrust and torque, together with the thrust induced pressure force on the hull, is fundamental to achieve high vessel control performance. The knowledge of the propeller thrust, either measured or estimated, could also allow the design of controllers for reducing power fluctuations and wear and tear in high sea state. Moreover, the performance monitoring is also useful This work has been supperted by the Norwegian Research Council. L. Pivano and T. A. Johansen are with Department of Engineering Cybernetics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway. E-mail: [luca.pivano,tor.arne.johansen]@itk.ntnu.no Ø. N. Smogeli is with Marine Cybernetics, NO-7075 Tiller, Norway. Email: [[email protected]]
for improving fault detection and thrust allocation in different propeller working conditions. These considerations motivate the development of schemes to estimate the propeller thrust because, in general, its measurement is not available. The estimated value of the thrust could be used for underwater vehicles for example, in observers for the estimation of the ocean current [7] and in adaptive schemes for the identification of the vehicle hydrodynamic drag [24]. Recently, observers for monitoring the propeller performance have been developed and included in new control designs for electrically driven propellers, see [3], [10], [18], [21], [25], and [28]. The problem of propeller thrust estimation has been treated in [33], where full-scale experimental results were provided in steady-state conditions, in waves, and for inclined inflow. The estimation was based on the propeller torque measurement and on a linear relation between thrust and torque. Experimental results were presented only for positive shaft speed and vessel speed. Steady-state thrust estimates can also be obtained from thrust and torque identity techniques [6] which assume the knowledge of the propeller torque, used to compute an equivalent open water advance ratio. This is combined with open-water propeller characteristics, corrected for scale effects, to obtain the thrust estimate. Thrust estimation has been also treated in [10], where the estimate was computed from the propeller torque obtained with a Kalman filter where a linear shaft friction torque was considered. The relation between thrust and torque involved an axial flow velocity model and requires the knowledge of the advance speed. The scheme was also highly sensitive to hydrodynamic and mechanical modeling errors. The performance was validated by simulations. An adaptive observer to estimate shaft speed and thrust was also designed for variable pitch propulsion systems in [3] and [18]. The observer was used for fault detection in the shaft speed control loop. A linear approximation of the propeller characteristics was utilized, therefore this approach could not guarantee accurate results in all four quadrant plane composed by the vessel speed and the propeller shaft speed. Moreover, the observer employed the vessel speed measurement. The contribution of this paper is the development of a fourquadrant thrust estimation scheme, extending the preliminary results described in [22] and [23]. The strength of the presented approach is that only measurements of the propeller shaft speed and the motor torque, normally available on ships, are utilized. Differently from [10], the advance speed which is very difficult to measure in real vessels, is assumed to be unknown.
IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY, VOL. XX, NO. Y, MONTH 2008
Our scheme is based on a robust nonlinear observer to estimate the propeller torque and the shaft speed, and on a mapping to compute the propeller thrust from the observer estimates. The observer is similar to the one introduced in [28], but the inclusion of a nonlinear friction term is a new contribution. Moreover, in order to analyze the effect of the measurement and friction modeling errors on the observer estimates, a Lyapunov based robustness analysis has been performed. The thrust is computed from the torque estimate through a mapping that involves the estimation of the advance ratio. The performance of the proposed scheme is demonstrated by extensive experiments carried out on an electrically driven fixed pitch propeller in a basin with close to open-water conditions, in waves and with a wake screen that scales the local flow down in order to simulate one of the effects of the interaction between the propeller and the vessel hull. The paper is organized as follows. The overall propulsion system is described in Section II. The experimental setup and instrumentation are presented in Section III. Modeling of the propeller shaft dynamics, thrust and torque, and the open-water characteristics is treated in Section IV. The thrust estimation scheme is described in Section V and experimental results are presented in Section VI. Finally, the conclusions are given in Section VII. II. OVERALL S YSTEM D ESCRIPTION
2
III. E XPERIMENTAL S ETUP AND I NSTRUMENTATION The tests were carried out at the Marine Cybernetics Laboratory, an experimental laboratory located at NTNU in Trondheim. The basin, 6.45 m wide, 40 m long and 1.5 m deep, is equipped with a 6DOF towing carriage that can reach a maximum speed of 2 m/s and with a wave generator able to generate waves up to 0.3 m. The tank dimensions may appear too small for accurate open-water and dynamic tests due to the influence of previous motions, presence of walls and free surface motion. The variance of the obtained results was found to be small. We employed a three phase brushless motor in combination with a drive equipped with a built-in torque controller and a build-in shaft speed controller. In this way we could choose to control the motor torque in order to obtain the desired motor torque or the shaft speed to obtain the desired ω. The motor was connected to the propeller shaft through a gear-box with ratio 1:1. The rig with motor, underwater housing, shaft and propeller was attached to the towing carriage in order to move the propeller through the water. The tests were performed on a fixed pitch propeller without duct and with geometric parameters given in Table R was used to interface I. The real-time system Opal RT-Lab R environment to the motor drive and the the Matlab/Simulink sensors. The shaft speed was measured on the motor shaft with a tachometer dynamo. The thrust and torque were measured with an inductive transducer and a strain gauge transducer placed on the propeller shaft, respectively. The measurement of the motor torque was furnished by the motor drive. All the signals were acquired at the frequency of 200 Hz. A sketch of the setup is shown in Fig. 2 and a picture of the propeller system is presented in Fig. 3.
Fig. 1. Block diagram of the propeller system and the thrust estimation scheme.
A block diagram that represents the propeller system and the scheme implemented to estimate the propeller thrust is shown in Fig. 1. The propeller is connected to the motor through a shaft and a gear box. The motor torque applied to the shaft is denoted Qm . The gear ratio is defined by Rgb = ωm /ω, where ωm is the motor shaft angular speed and ω is the propeller angular speed. The value of ω is particularly influenced by the load, represented by the propeller torque Qp , due to the rotation of the blades in the water. The output of the system is the thrust Tp produced by the propeller. The thrust estimation scheme includes a nonlinear observer ˆ p of the propeller load torque that computes the estimate Q and the estimate ω ˆ of the shaft speed. The observer uses the measurements of the motor torque Qm and the propeller shaft speed ω. An estimate Tˆp of the propeller thrust is computed ˆ p and ω using the observer estimates Q ˆ through a mapping f.
Fig. 2.
Sketch of the experimental setup.
TABLE I P1362 PROPELLER GEOMETRICAL PARAMETERS
Parameter D Z P/D Ae /A0
Value 0.25 m 4 1.0 0.55
Description Propeller diameter Number of blades Pitch ratio P/D Expanded blade area ratio
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With Rgb = 1 and defining z = Qm − Qp , θ is computed over a time-series of N samples as θ = arg min
N
2
|zi − Qf (ωi ) − Jm ω˙ i | .
(4)
i=1
where the subscript i indicates the i-th sample (see for examples [9]). The parameters obtained are shown in Table II. TABLE II F RICTION MODEL PARAMETERS AND SHAFT MOMENT OF INERTIA .
IV. P ROPELLER M ODELING
A. Propeller shaft dynamics The shaft dynamics is derived by considering the motor connected to the propeller through a rigid shaft and a gearbox with gear ratio Rgb , as shown in the block diagram of Fig. 1. The shaft is considered affected by a friction torque denoted Qf (ω), which is assumed to depend only on the shaft speed. The shaft dynamics can be written as Jm ω˙ = Rgb Qm − Qp − Qf (ω),
(1)
where Jm is total moment of inertia including the shaft, the gear box and the propeller. The friction torque has been modeled as ω
+ kf2 ω + kf3 arctan(kf4 ω), (2) where the Coulomb effect, usually written as a sign(ω), has been replaced by the function π2 arctan( ω ) with a small positive in order to avoid the singularity for ω = 0. The remaining terms in (2) represent a linear and a nonlinear viscous effect. All the coefficients kfi are constant and positive. The static friction model (2) is able to approximate the friction torques experienced in practice (see [1], [16] and [22]). Qf (ω) = kf1 arctan
1) Shaft moment of inertia and friction torque identification: To identify the friction torque and the shaft moment of inertia in (1), we ran tests with different motor torque profiles and various towing carriage speeds. From the measurement of the propeller angular speed, the motor torque and the propeller torque, and computing the derivative of ω with the necessary filtering, we identified the parameters kfi of the friction torque model (2) and the shaft moment of inertia Jm . The parameters kfi and Jm can be grouped in the vector T θ = kf1 kf2 kf3 kf4 Jm . (3)
Value 6.07 · 10−3 3.97 · 10−1 9.28 · 10−3
Parameter kf3 [kg · m2 /s] kf4 [−] [−]
Value 6.61 · 10−3 8.94 · 10−2 1 · 10−3
Figure 4 shows the friction torque computed from measurements and the identified model. The friction exhibits a nonlinear behavior and is affected by the temperature in the gears, bearings and oil. The friction presents also a hysteresis effect but its influence is not very significant and it has been neglected. For the tested propeller system, the losses due to the friction torque are quite high compared to a full scale propeller, where they are usually less significant. 3 2
Qf [N m]
Fig. 3. Propeller open-water configuration (main picture) and wake screen (small picture).
Parameter Jm [kg · m2 ] kf1 [kg · m2 /s] kf2 [kg · m2 /s]
1 0 -1 -2 -3 -80
Measured Qf Qf model -60
-40
-20
0
20
40
60
80
ω [rad/s] Fig. 4. Friction torque: computed from measurements and the identified nonlinear static model.
B. Propeller thrust and torque
Fig. 5. Definition of the advance speed ua and vessel speed U and the undisturbed flow speed uu .
IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY, VOL. XX, NO. Y, MONTH 2008
Modeling the thrust and torque produced by a propeller is a complicated task, since it is difficult to develop a finitedimensional analytical model from the laws of physics. This is mainly due to the difficulty in modelling the flow dynamics, especially when the flow is not uniform [1], [4], [5], [12], [16], [20]. The thrust and torque depend also on the propeller geometric parameters (i.e. propeller diameter, pitch angle, etc.), the non-dimensional parameters advance ratio J and Reynolds number, the propeller submergence and environmental state (waves, currents, etc.). A common practice is the use of simplified models which are chosen based on the propeller application. See for example [8], [21] and [28] and the references therein. 1) Open-Water propeller characteristics: Neglecting the effect of waves and marine currents, and assuming a deeply submerged fixed pitch propeller, the thrust and torque are usually represented in nondimensional form. One form is represented by the standard open-water coefficients KT and KQ , given as functions of the advance ratio J. The term openwater refers to the case where the propeller is tested without the presence of a vessel hull. The coefficients KT and KQ are computed from [6] as 4π 2 , (5) KT = T p ρ |ω| ωD4 KQ = Qp
4π 2 , ρ |ω| ωD5
(6)
where ρ is the density of the water and D is the propeller diameter. The advance ratio is computed as 2πua , (7) ωD where ua is the advance speed, i.e. the water inflow velocity to the propeller. The KT and KQ curves are measured for a range of propeller advance numbers J, usually in a cavitation tunnel or a towing tank [14]. When the propeller is working in water that has been disturbed by the passage of the hull, it is no longer advancing relatively to the water at the speed of the ship U, but at some different speed ua . The advance speed is very difficult to measure and an estimate of ua is usually computed using the steady-state relation J=
ua = (1 − w)U,
(8)
where w is the wake fraction number, often identified from experimental tests (see e.g. [17]). Figure 5 shows a sketch of a vessel with the velocities involved. The surge vessel speed U is relative to the earth while ua is the longitudinal water speed relative to the propeller disc. The undisturbed water velocity uu has the same magnitude as the vessel speed but with opposite direction. A measure of the propeller performance is the open-water efficiency, which is defined as the ratio of the produced to the consumed power by the propeller. The propeller efficiency is usually plotted for positive values of J and is computed from (5), (6) and (7) as η=
u a KT JKT u a Tp = = . ωQp ωDKQ 2πKQ
(9)
4
The curves KT and KQ are usually employed in the first and in the third quadrant of the plane composed by ua and ω, and they are not defined for ω = 0. For propellers operating in the whole plane (ua ,ω), four-quadrant open-water characteristics are normally utilized [6]. The four-quadrant coefficients CT and CQ are plotted as functions of the advance angle β. The value of β is computed with the four quadrant inverse tangent function as β = arctan2 (ua , 0.7Rω) ,
(10)
where R is the propeller disc radius. The four-quadrant coefficients are calculated from [6] as CT = CQ =
Tp , 1 2 2 ρVr A0
(11)
Qp , 1 2 2 ρVr A0 D
(12)
where A0 is the propeller disc area and Vr is the relative advance velocity: Vr2 = u2a + (0.7Rω)2 .
(13)
2) Measured open-water characteristics: To measure the open-water propeller characteristics, we performed tests at different values of the advance ratio J. To obtain the desired shaft speed ω, the built-in speed controller of the drive was used. In our setup, the housing that contains gear and measurement devices does not create a significant wake and the advance speed ua has been considered equal to the towing carriage speed U. This yields a wake fraction number w equal to zero. The standard propeller characteristics are plotted in Fig. 6. In Fig. 7 a sample of filtered data used to derive the KT and KQ curves are plotted. For positive ua and ω, the inflow to the propeller is uniform and the thrust and torque are quite steady. When the advance speed becomes negative, the propeller tries to reverse the inlet flow and a recirculation zone (often called a ring vortex) occurs [30]. This is due to the interaction between the inlet flow and the reversed flow. The flow then becomes unsteady and can cause oscillations in the propeller thrust and torque. For negative values of J, KT and KQ were computed from the average of the measured thrust and torque. From Fig. 6, it can be noticed that the tested propeller is not symmetric in the thrust production with respect to the shaft speed. For positive values of ω the efficiency is higher because the propeller was designed to work mainly at forward vessel speed. Figure 8 shows the four-quadrant propeller characteristics and an approximation computed with a 25th order Fourier series, commonly adopted for the CT and CQ curves [6] . 3) Torque model for the observer: In order to estimate the propeller torque with an observer, a dynamic model for Qp is developed. The propeller torque Qp is treated as a timevarying parameter and modeled as a first order process with a positive time constant τ1 , driven by a bounded noise w1 as in [22], [23] and [26]: 1 Q˙ p = − Qp + w1 . (14) τ1
IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY, VOL. XX, NO. Y, MONTH 2008
5
2.5
1.5
u 0
10KQ KT η
u >0, ω>0
a
a
Fourier CT Fourier −10CQ meas CT meas −10CQ
2
1
1.5 1
CT , −10CQ
0.5
(a) 0 1.5
u >0, ω −kf2 + αl22κ + 2μκ our analysis to values of J in the range [−1.5, 1.1]. However, then the system (19) is input-to-state stable (ISS). when the shaft speed is reversed, the propeller works for a Proof: See Appendix. short time outside the range of J that has been considered. Remark 2: For the observer considered, there always exist In this condition, both thrust and torque are small since the parameter and gain values that can be chosen according to the shaft speed is small, and the approximation error of GQT (J) above criteria. does not affect the estimation significantly. This is shown in The ISS property of the observer error dynamics provides the experimental results reported in Section VI. ˆp the robustness of the observer against noise and modeling The estimation of J is performed using the estimates Q ˆ Q , an estimate of KQ as errors. The observer errors, and thus the torque and shaft and ω ˆ . From (6), we can compute K speed estimates, remain bounded for any initial conditions 2 regardless of the values of the measurement errors, the noise ˆQ = Q ˆ p 4π , ω K ˆ = 0. (27) ρˆ ω 2 D5 w1 in the propeller torque model and the difference between the friction torque model and the actual friction. In particular, The value of K ˆ Q has been limited by the upper bound the observer robustness against friction torque modeling errors K (J ) and lower bound K (J Q min Q max ) where [Jmin , Jmax ] is very important. The shaft friction torque may depend upon is the range of J. In this way we handle also the case when variables which are not directly accounted for in the model, ω ˆ = 0. An estimate Jˆ of the advance ratio can be derived by like temperature and bearing lubrication. ˆ Q computed inverting the KQ curve and using the value of K with (27). It can be noticed in part (a) of Figures 9 and 10, B. Thrust and torque relationship that the KQ curve is not invertible in the whole range of As stated above it is difficult to derive accurate models J considered. For this reason, the J axis has been divided for the thrust and torque, especially when the inflow to in three zones. In zones 1 and 3 the KQ curve is invertible the propeller is not uniform. For example, [1], [11] and and an accurate estimate of J can be computed. In zone 2, [20] experimentally demonstrated the need of including the Jˆ has been approximated with zero to ensure correct thrust
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ˆ for positive and Plots of the gains GQT (J) and GQT (J), ˆp negative values of ω, are shown Figures 9 and 10. If Q ˆ and ω ˆ are accurate, outside zone 2, GQT (J) approximates accurately GQT (J) and the estimated thrust is precise. In ˆ is equal to GQT (0) and, due to the propeller zone 2, GQT (J) ˆ and GQT (J) characteristics, the difference between GQT (J) is not of significant magnitude. In the border of zone 2, ˆ has been joined smoothly to GQT (0) in order to GQT (J) avoid sharp variation on the thrust estimate. For the tested propeller, the value of J is limited to the range [-1.5,1.1] for ω ≥ 0 and to the range [-1.5,0.9] for ω < 0. The maximum ˆ and GQT (J) is about 8% for relative error between GQT (J) ω ≥ 0 and 13% for ω < 0.
J -1.5
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0.25
Zone 1
K Q (ω < 0)
0.2
Zone 2
Zone 3
0.15 0.1 0.05
(a)
0
25
T p /Q p (ω < 0)
estimation when the advance speed is zero, i.e. the vessel is at rest and not subjected to current. This approximation introduces an error on the estimate which is computed as ˆ p GQT (J)| ˆ ωˆ ≥0 Q ω ˆ≥0 Tˆp = (28) ˆ ˆ ωˆ 4.1 · 10−2 + 3.5 · 10−12 l2 , 0 < l2 < 5 · 1020 ,
Remark 3: On full scale vessels, the open-water characteristics obtained in a model scale is expected to be corrected for scale effects [13]. Remark 4: If the propeller open-water characteristics are not available, Computational Fluid Dynamics (CFD) techniques can help to derive it from the 3D drawing of the propeller, see for example [19] and references therein. Remark 5: The thrust and torque relationship is not derived from the four-quadrant propeller characteristics because it is ˆp not possible to estimate the advance angle β only from Q and ω ˆ , since we cannot compute the coefficient CQ without knowing ua . This makes this parameterization difficult to use ˆp. ˆ and Q while we can estimate J from KQ , computed using ω
which practically allows us to choose l1 and l2 freely. Considering the fact that we want the observer dynamics faster than the system dynamics and at the same time, too high gains can produce oscillatory estimates due to the measurement noise on the shaft speed, l1 and l2 were chosen as a trade-off between the two opposite requirements. The time constant was obtained from a sensitivity analysis on the observer estimation errors with respect to τ1 . Running the observer with l1 = 3 kg · m2 /s, l2 = 80 kg · m2 /s2 , on data acquired over more than 1 h of tests carried out in open-water conditions at different advance speeds and shaft speeds, we derived Fig. 12. The graph shows the root mean square error (RMSE) between the observer estimates and the
IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY, VOL. XX, NO. Y, MONTH 2008
2
ω ˜ ˜p Q
RSME
1.5
60
ω [rad/s]
20 0 -20 -40
(a)
-60 4 2 0 -2 -4
(b)
6 measured Qp ˆp Q
1
4
0
0.01
0.1
1
10
Q p [N m]
0.5
100
τ1 [s]
B. Open water tests A series of tests to validate the proposed scheme have been carried out in open-water conditions. The propeller was submerged at h/R = 4, where h is the propeller submergence, in order to avoid losses due to ventilation. Figures 13 and 14 show data from an experiment where both the advance speed and the shaft speed had a trapezoidal form. Figures 15 and 16 show data from a test with sinusoidal advance and shaft speed. The shaft speed and propeller torque estimates are plotted in part (a) and (c) of Figures 13 and 15, respectively. Both estimates are accurate and almost indistinguishable from the measurements. The observer estimation errors are plotted in part (b) and (d) of Figures 13 and 15. The advance speed is plotted in part (e) of the same figures. Figures 14 and 16 show the measured and estimated thrust from the same tests. The estimated Tˆp , shown in part (a), obtained with the proposed method reproduces quite well the measurements in all the quadrants. Part (e) of the same figures shows the advance angle β. The estimate Tˆp is compared with the estimate TˆpCT , shown in part (c), computed using the measured four-quadrant propeller characteristics CT , introduced in Section IV, with ua = U. This estimate is not very accurate, especially in the 2nd and 4th quadrant where the inflow to the propeller can be irregular. The proposed scheme furnish a more accurate thrust estimate since it can sense the effect of the flow variation through the propeller torque estimate. The thrust estimation errors are shown in (b) and (d) of Figures 14 and 16. C. Wake screen test Tests with a wake screen, shown in Fig. 3, were performed to simulate one of the effects of the hull on the propeller inflow. The wake screen created an uniform loss of speed. This does not represent entirely the effect of the hull because
ˆ p [Nm] Qp − Q
Observer estimation errors for different values of τ1 .
2 0 -2 -4
(c)
-6 1 0.5 0 -0.5 -1
(d)
2
u a [m/s]
Fig. 12.
measured ω ω ˆ
40
ω−ω ˆ [rad/s]
measurements. The value of the time constant has been varied between 0.01 and 100. For τ1 ≥ 1, the accuracy of the estimates is practically the same, while for smaller values the estimate are less precise. For small values of τ1 , the torque estimation error decreases when the shaft speed estimation error diminishes. This allows us to choose the time constant based on the speed error, since the torque measurements is not available in real cases. The observer parameters used in the experiments are l1 = 3 kg · m2 /s, l2 = 80 kg · m2 /s2 and τ1 = 10 s.
8
(e)
1 0 -1 -2
795
805
815
825
835
Time [s] Fig. 13.
Data from the first open-water test.
the propeller inflow is not usually uniform. The propeller was submerged at h/R = 4. Figure 17 shows the results from a test where both the advance speed and the shaft speed had a trapezoidal form. As for the open-water experiments, the estimates provided by the observer, shown in (a) and (b), are very accurate. In (d) and (e) of the same plot, the estimate Tˆp obtained with the proposed method is compared with the estimate TˆpCT , computed using the measured four-quadrant propeller characteristic CT . The advance speed is computed with (8), where the value of the wake fraction w has been identified from tests performed in steady-state conditions. For positive towing carriage speed U , shown in (c), the experimentally found value was w = 0.3. For negative towing carriage speed, the inlet water flow to the propeller was not affected by the grid, placed upstream of the propeller, and the wake fraction number was zero. The estimate Tˆp is quite accurate also in this experiment while the estimate TˆpCT , as for the open-water tests, is accurate only
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40 0 -40
0 -20
(a)
-60
ω−ω ˆ [rad/s]
Tp − Tˆp [N]
-120 60 30 0 -30 -60
20
-40
(a)
-80
measured ω ω ˆ
40
ω [rad/s]
80
T p [N ]
60
measured Tp Tˆp
120
9
(b) measured Tp Tˆp
120
4 2 0 -2 -4 6
(b) measured Qp ˆp Q
4
CT
Q p [N m]
T p [N ]
80 40 0 -40
ˆ p [Nm] Qp − Q
Tp − TˆpCT [N]
-120 60 30 0 -30 -60 360
(d)
β [deg]
rd
3
180 nd
2
(e)
st
1
0
795
805
815
-2
(c)
-6 1 0.5 0 -0.5 -1 2
u a [m/s]
th
4
270
90
0
-4
(c)
-80
2
835
(e)
1 0 -1 -2 1200
825
(d)
1210
1220
1230
1240
1250
1260
1270
1280
Time [s]
Time [s] Fig. 15. Fig. 14.
Data from the second open-water test.
Data from the first open-water test.
when ω and the ua have the same sign. D. Test in waves Tests in waves were carried-out in order to validate the estimation scheme with a periodic propeller inflow and with large losses due to ventilation. Figure 18 shows the result of a test performed in regular waves with amplitude 0.05 m (equivalent to 0.2 D) and frequency 0.69 Hz. The propeller was also moved along its vertical axis with a sinusoidal motion. This was done to simulate the motion that a propeller may experience in rough sea conditions. This test does not reproduce entirely rough sea conditions but it may be a valid indication of the performance of the proposed method when operating in off-design conditions. The towing carriage was kept at rest (U = 0), but the advance speed was still not zero since the waves create an inflow to the propeller. Part (c) of Fig. 18 shows the propeller vertical displacement d. The propeller shaft speed, depicted in part (a), has been kept constant at 38 rad/s. A drop of thrust and torque occurred when the propeller rotated close to the water surface, since
the load decreased due to ventilation. The small oscillations of thrust and torque were due to waves that created a periodic additional axial velocity component that varied with depth and time across the propeller plane. Both phenomena were well reproduced by the torque estimate, as shown in part (b) of Fig. 18. The thrust estimate Tˆp obtained from the proposed method is depicted in Fig. 18 (d) and is compared with TˆpCT , the thrust computed directly from the CT characteristic. The proposed method produced a satisfactory estimate and both the oscillations and the drop of thrust were properly captured. The estimate TˆpCT has been computed assuming ua = 0, the best guess we could make since U = 0. Since ω was constant and ua = 0, the estimate TˆpCT was constant and could not capture the thrust variations due to waves. VII. C ONCLUSION A thrust estimation scheme for a marine propeller has been developed and experimentally tested on an electrically driven propeller. Tests were performed in open-water conditions, with a wake screen to simulate one of the effects of the hull on the propeller inflow and in waves with vertical propeller motion in
IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY, VOL. XX, NO. Y, MONTH 2008
ω [rad/s]
measured Tp Tˆp
120
T p [N ]
80 40 0
(a)
-80
Tp − Tˆp [N]
-120 60 30 0 -30 -60
U [m/s]
(b)
measured Tp TˆpCT
120
T p [N ]
80
Tp [N ]
40 0
-120 60 30 0 -30 -60
Tp [N ]
(c)
-80
Tp − TˆpCT [N]
measured ω ω ˆ
(a)
(b)
(d)
360 4th
β [deg]
270 rd
3
180 2nd
90
(e)
st
1
0 1200
1210
1220
1230
1240
0 -1
(c)
120 80 40 0 -40 -80 -120 120 80 40 0 -40 -80 -120 360 270 180 90 0 161
1250
1260
1270
1280
measured Qp ˆp Q
1
-2
-40
β [deg]
60 40 20 0 -20 -40 -60 6 4 2 0 -2 -4 -6 2
Qp [N m]
-40
10
(d)
measured Tp Tˆp
(e)
measured Tp TˆpCT
4th 3rd 2nd
(f)
1st
171
181
Time [s]
Time [s] Fig. 16.
Data from the second open-water test.
order to reproduce the motion that a propeller may experience in rough sea conditions. The robustness of the scheme with respect to modeling and measurement errors was demonstrated with the use of Lyapunov theory and corroborated by experimental results. The scheme involved a nonlinear observer to estimate the propeller torque and the shaft speed using only the measurements of the motor torque and the propeller shaft speed. The thrust estimate was computed from the estimated propeller torque and shaft speed involving the estimation of the advance ratio J. This only required knowledge of the standard propeller characteristics. The thrust obtained from the proposed method was compared with the thrust computed from the four-quadrant propeller characteristics showing improved accuracy in the estimates. The thrust estimation scheme can be implemented also for ducted propellers, where the standard propeller characteristics are slightly different compared to a propeller without a duct. Although the presented results concern tests carried out on an electrically driven propeller, the scheme could be applied
Fig. 17.
Data from the wake screen experiment.
also to propellers driven by diesel motors where the motor torque can be measured with strain gauges on the motor shaft [2] or by measuring the fuel index [3]. The presented results are promising for the use of such a thrust estimation scheme in high performance propeller controllers. VIII. ACKNOWLEDGMENTS The authors acknowledge with gratitude Professor Thor Inge Fossen for valuable suggestions and discussions. Matthias Schellhase is gratefully acknowledged for the help given during the experimental tests. The Research Council of Norway is acknowledged as the main sponsor of this project. A PPENDIX The appendix presents the proof of Proposition 1. First we consider the input u, defined in (18), equal to zero ∀t and later we investigate its effect on the error dynamics. Taking the Lyapunov function candidate V := 12 e˜T P1 e˜, where P1 = P1T > 0 and
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(a)
ω [rad/s]
60
V˙ ≤
measured ω ω ˆ
40
−p12 g1 (˜ e1 , ω ˆ )˜ e2 .
Qp [Nm]
4
(33)
on the last term of (33) obtaining (b)
2
measured Qp ˆp Q
−p12 g1 (˜ e1 , ω ˆ )˜ e2
80
(d) 40 measured Tp Tˆp
(e) 40 measured Tp TˆpCT
Data from the experiment with waves.
P1 =
p11 p12
p12 p22
With this choice of l1 , we observe from (36) that p22 > 0 and the inequality (38) certainly holds if 2 μ< . (40) Jm
,
(29)
we can compute its time derivative along the trajectory of (19) obtaining
p12 V˙ = − J1m p11 (kf2 + l1 ) − p12 l2 e˜21 − pτ22 e˜22 + J m
1 − pJ12 (kf2 + l1 ) + pτ12 + pJ11 − p22 l2 e˜1 e˜2 m 1 m −p12 ψ(˜ e1 , ω ˆ )˜ e2 − p11 ψ(˜ e1 , ω ˆ )˜ e1 . (30) ˆ ) we can subtract the linear From the nonlinearity ψ(˜ e1 , ω function α˜ e1 , where α is constant and satisfies A1, such that g1 (˜ e1 , ω ˆ ) = ψ(˜ e1 , ω ˆ ) − α˜ e1 .
(31)
Since the graph of ψ(˜ e1 , ω ˆ ) belongs to the sector [0, α], the ˆ ) belongs to the sector [−α, α], i.e. graph of g(˜ e1 , ω 2
ω : [g(˜ e1 , ω ˆ )] < ∀˜ e1 , ∀ˆ
(34)
Choosing p11 = p12 α2 κJm with κ > 0, the inequality (37) is satisfied for 1 l2 . (39) l1 > −kf2 + 2 + α κ 2μκ
20
Time [s]
2 p12 e1 , ω ˆ )] 2μ [g1 (˜ p12 2 2 ˜1 . 2μ α e
the cross-term in (35) is cancelled. To obtain a negative definite V˙ the following are needed: 1 p12 2 α > 0, p11 (kf2 + l1 ) − p12 l2 − (37) Jm 2μ 1 p22 μ + p12 − > 0. (38) τ1 Jm 2
80
0 10
≤ μ p212 e˜22 + ≤ μ p212 e˜22 +
Using (34) in (33) we attain
12 V˙ ≤ − J1m p11 (kf2 + l1 ) − p12 l2 − p2μ α2 e˜21
p12 p11 − pJ12 (k + l ) + + − p l + p α e˜1 e˜2 f 1 22 2 12 2 τ J m
1 m − pτ22 + pJ12 − μ p212 e˜22 . 1 m (35) Selecting l2 > 0 and p22 such that
1 p12 p12 p11 (kf2 + l1 ) + + + p12 α , (36) p22 = l2 Jm τ1 Jm
(c)
0
Tp [N]
Choosing p12 > 0, we can use Young’s inequality 1 −2xy ≤ μx2 + y 2 ∀μ > 0, μ
0 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Tp [N]
d [m]
1 e˜22 p11 (kf2 + l1 ) − p12 l2 e˜21 − pτ22 + pJ12 1 m Jm − pJ12 (kf2 + l1 ) + pτ12 + pJ11 − p22 l2 + p12 α e˜1 e˜2 m 1 m
−
20 0
Fig. 18.
11
α2 e˜21 .
(32)
Substituting (31) in (30) and recalling that, since p11 > 0, p11 ψ(˜ e1 , ω ˆ )˜ e1 > 0 we get
Combining (39) and (36), we get p22 > l2pJ11m . This yields
p11 p11 2 κJ α m P1 > . (41) p11 p11 α2 κJm
l2 Jm
If l2 < α κ Jm then P1 is positive definite. Choosing the observer gains according to A4 and A5 of Proposition 1, the derivative of the Lyapunov function candidate is negative definite since 2 (42) e , V˙ ≤ − min{q1 , q2 } ˜ 4 2
2
where
1 p12 2 α , p11 (kf2 + l1 ) − p12 l2 − Jm 2μ 1 p22 μ q2 = + p12 − . τ1 Jm 2
q1 =
(43) (44)
The observer error dynamics, with u = 0 ∀t, is thus globally exponentially stable (GES). When the input u is different from zero for some t, the term 2˜ eT P1 B1 u must be added to the
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derivative of the Lyapunov function in (42): 2
≤ − min{q1 , q2 } ˜ e2 + 2˜ eT P1 B1 u 2 ≤ − min{q1 , q2 } ˜ e2 + 2 e˜T P1 B1 u2 2 ≤ − min{q1 , q2 } ˜ e2 + 2 e˜T 2 P1 2 B1 2 u2 . (45) With 0 < θ < 1, we obtain V˙
2
e2 + P1 2 B1 2 ˜ e2 u2 ≤ − min{q1 , q2 } ˜ 2 2 ≤ −(1 − θ) min{q1 , q2 } ˜ e2 − θ min{q1 , q2 } ˜ e2 + P1 2 B1 2 ˜ e2 u2 . (46) For any ˜ e2 such that V˙
˜ e2 ≥ ρ(u2 ), where ρ(u2 ) =
P1 2 B1 2 u2 θ min{q1 , q2 }
(47) (48)
is a (linear) class K function, we obtain 2 V˙ ≤ −(1 − θ) min{q1 , q2 } ˜ e2 ≤ 0.
(49)
Since V is positive definite and radially unbounded, from Theorem 4.19 in [15], the system (19) is ISS. Furthermore, the observer erroris uniformly ultimately bounded (UUB) by ρ supt>t0 (u2 ) . R EFERENCES [1] R. Bachmayer, L. L. Whitcomb, and M. A. Grosenbaugh, “An accurate four-quadrant nonlinear dynamical model for marine thrusters: Theory and experimental validation,” IEEE Journal of Oceanic Engineering, vol. 25, no. 1, pp. 146–159, January 2000. [2] M. Blanke, “Ship propulsion losses related to automatic steering and prime mover control.” Ph.D. dissertation, Technical University of Denmark, 1981. [3] M. Blanke, R. Izadi-Zamanabadi, and T. F. Lootsma, “Fault monitoring and reconfigurable control for ships propulsion plant,” Journal of Adaptive Control and Signal Processing, vol. 12, pp. 671–688, 1998. [4] M. Blanke, K. Lindegaard, and T. I. Fossen, “Dynamic model for thrust generation of marine propellers,” in 5th IFAC Conference of Manoeuvring and Control of Marine craft (MCMC), Aalborg, Denmark, 2000, pp. 363–368. [5] J. P. Breslin and P. Andersen, Hydrodynamics of Ship Propellers. Cambridge University Press, 1994. [6] J. S. Carlton, Marine Propellers and Propulsion. Oxford, U.K.: Butterworth-Heinemann Ltd., 1994. [7] T. I. Fossen, Marine Control Systems - Guidance, Navigation, and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, Trondheim, Norway, 2002. [8] T. I. Fossen and M. Blanke, “Nonlinear output feedback control of underwater vehicle propellers using feedback from estimated axial flow velocity,” IEEE Journal of Oceanic Engineering, vol. 25, no. 2, pp. 241–255, April 2000. [9] P. Gill, W. Murray, and M. Wright, Practical Optimization. London, Academic Press, 1981. [10] C. Guibert, E. Foulon, N. A¨ıt-Ahmed, and L. Loron, “Thrust control of electric marine thrusters,” in 31nd Annual Conference of IEEE Industrial Electronics Society. IECON 2005, Raleigh, North Carolina, USA, 6-10 November 2005. [11] A. J. Healey, S. M. Rock, S. Cody, D. Miles, and J. P. Brown, “Toward an improved understanding of thruster dynamics for underwater vehicles,” Symp. Autonomous Underwater Vehicle Technology, Boston MA, pp. 340–352, 1994. [12] ——, “Toward an improved understanding of thruster dynamics for underwater vehicles,” IEEE Journal of Oceanic Engineering, vol. 20, no. 4, pp. 354–61, October 1995. [13] ITTC, Recommended Procedures - Performance, Propulsion 1978 ITTC Performance Prediction Method - 7.5-02-03-01.4, International Towing Tank Conference, 1999.
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[14] ——, Recommended Procedures - Testing and Extrapolation Methods. Propulsion, Propulsor, Open Water Test, International Towing Tank Conference, 2002. [15] H. K. Khalil, Nonlinear Systems, 3rd ed. Prentice Hall, 2000. [16] J. Kim and W. K. Chung, “Accurate and practical thruster modeling for underwater vehicles,” IEEE Journal of Ocean Engineering, vol. 33, no. 5-6, pp. 566–586, April 2006. [17] E. V. Lewis, Principles of Naval Architecture Vol II: Resistance, Propulsion and Vibration, 3rd ed. New York: Society of Naval Architects and Marine Engineers, 1988. [18] T. F. Lootsma, “Observer-based faut detection and isolation for nonlinear systems,” Ph.D. dissertation, Aalborg University, Denmark, 2001. [19] J. Mart´ınez-Calle, L. Balbona-Calvo, J. Gonz´alez-P´erez, and E. BlancoMarigorta, “An open water numerical model for a marine propeller: A comparison with experimental data,” in ASME FEDSM Joint USEuropean Fluids Engineering Summer Conference, Montreal, Canada, 2002. [20] L. Pivano, T. I. Fossen, and T. A. Johansen, “Nonlinear model identification of a marine propeller over four-quadrant operations,” in 14th IFAC Symposium on System Identification, SYSID, Newcastle, Australia, 2006. [21] L. Pivano, T. A. Johansen, Ø. N. Smogeli, and T. I. Fossen, “Nonlinear Thrust Controller for Marine Propellers in Four-Quadrant Operations,” in 26th American Control Conference (ACC07), New York, USA, July 2007. [22] L. Pivano, Ø. N. Smogeli, T. A. Johansen, and T. I. Fossen, “Experimental Validation of a Marine Propeller Thrust Estimation Scheme,” in 7th IFAC Conference on Manoeuvring and Control of Marine Craft (MCMC), Lisbon, Portugal, September 2006. [23] ——, “Marine propeller thrust estimation in four-quadrant operations,” in 45th IEEE Conference on Decision and Control, San Diego, CA, USA, 13-15 December 2006. [24] D. A. Smallwood and L. L. Whitcomb, “Model Based Dynamic Positioning of Underwater Robotic Vehicles: Theory and Experiment,” IEEE Journal of Oceanic Engineering, vol. 29, no. 1, pp. 169–186, January 2004. [25] Ø. N. Smogeli, “Control of Marine Propellers: From Normal to Extreme Conditions,” Ph.D. dissertation, Department of Marine Technology, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, September 2006. [26] Ø. N. Smogeli, J. Hansen, A. J. Sørensen, and T. A. Johansen, “Antispin control for marine propulsion systems,” in 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, 14-17 December 2004. [27] Ø. N. Smogeli, E. Ruth, and A. J. Sorensen, “Experimental validation of power and torque thruster control,” in IEEE 13th Mediterranean Conference on Control and Automation (MED’05), Cyprus, June 2005, pp. 1506– 1511. [28] Ø. N. Smogeli, A. J. Sørensen, and T. I. Fossen, “Design of a hybrid power/torque thruster controller with loss estimation,” in IFAC Conference on Control Applications in Marine Systems (CAMS’04), Ancona, Italy, 2004. [29] Ø. N. Smogeli, A. J. Sørensen, and K. J. Minsaas, “The Concept of Anti-Spin Thruster Control,” To appear in Control Engineering Practice (CEP), 2007. [30] M. Vysohlid and K. Mahesh, “Large-eddy simulation of propeller crashback,” in 57th Annual Meeting of the Division of Fluid Dynamics, Seattle, Washington, 21-23 November 2004. [31] L. L. Whitcomb and D. Yoerger, “Developement, comparison, and preliminary experimental validation of nonlinear dynamic thruster models,” IEEE Journal of Oceanic Engineering, vol. 24, no. 4, pp. 481–494, Oct. 1999. [32] D. R. Yoerger, J. G. Cooke, and J. E. Slotine, “The influence of thruster dynamics on underwater vehicle behavior and their incorporation into control system design,” IEEE Journal of Oceanic Engineering, vol. 15, no. 3, pp. 167–178, June 1990. [33] V. B. Zhinkin, “Determination of the screw propeller thrust when the torque or shaft power is known,” in Fourth international symposium on practical design of ships and mobile units, Bulgaria, 23-38 October 1989.
1
A Four-Quadrant Thrust Estimation Scheme for Marine Propellers: Theory and Experiments Luca Pivano, Tor Arne Johansen and Øyvind N. Smogeli
Abstract— A thrust estimation scheme for marine propellers that can operate in the full four-quadrant range of the propeller shaft speed and the vessel speed has been developed. The scheme is formed by a nonlinear observer to estimate the propeller torque and the propeller shaft speed and by a mapping to compute the thrust from the observer estimates. The mapping includes the estimation of the propeller advance ratio. The advance speed is assumed to be unknown, and only measurements of shaft speed and motor torque have been used. The robustness of the scheme is demonstrated by Lyapunov theory. The proposed method is experimentally tested on an electrically driven fixed pitch propeller in open-water conditions, in waves and with a wake screen that scales the local flow down in order to simulate one of the effects of the interaction between the propeller and the vessel hull. Index Terms— estimation, nonlinear, marine propulsion.
I. I NTRODUCTION In marine guidance, navigation and control (GNC) systems, the low level thruster controllers have traditionally received less attention compared to the guidance system and the highlevel plant control. In the design of Dynamic Positioning (DP), thruster assisted Position Mooring (PM) and autopilot systems, for example, much effort has been put into the high-level control schemes and the propeller dynamics has often been neglected. More recently, also the issue of thruster dynamics and control has received more attention. For recent references, see for example [1], [4], [8], [12], [20], [27], [29], [31], [32] and the references therein. The main difficulties in the design of effective propeller controllers lie in the modeling of the propeller’s dynamics and in the problem of measuring the environmental state. When a ship performs a marine operation, propellers are often affected by thrust losses due to cross flow, ventilation, in-and-out-of water effects, wave-induced water velocities, interaction between the vessel hull and the propeller and between propellers. Propellers may thus work far from ideal conditions therefore, knowledge of the propeller thrust and torque, together with the thrust induced pressure force on the hull, is fundamental to achieve high vessel control performance. The knowledge of the propeller thrust, either measured or estimated, could also allow the design of controllers for reducing power fluctuations and wear and tear in high sea state. Moreover, the performance monitoring is also useful This work has been supperted by the Norwegian Research Council. L. Pivano and T. A. Johansen are with Department of Engineering Cybernetics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway. E-mail: [luca.pivano,tor.arne.johansen]@itk.ntnu.no Ø. N. Smogeli is with Marine Cybernetics, NO-7075 Tiller, Norway. Email: [[email protected]]
for improving fault detection and thrust allocation in different propeller working conditions. These considerations motivate the development of schemes to estimate the propeller thrust because, in general, its measurement is not available. The estimated value of the thrust could be used for underwater vehicles for example, in observers for the estimation of the ocean current [7] and in adaptive schemes for the identification of the vehicle hydrodynamic drag [24]. Recently, observers for monitoring the propeller performance have been developed and included in new control designs for electrically driven propellers, see [3], [10], [18], [21], [25], and [28]. The problem of propeller thrust estimation has been treated in [33], where full-scale experimental results were provided in steady-state conditions, in waves, and for inclined inflow. The estimation was based on the propeller torque measurement and on a linear relation between thrust and torque. Experimental results were presented only for positive shaft speed and vessel speed. Steady-state thrust estimates can also be obtained from thrust and torque identity techniques [6] which assume the knowledge of the propeller torque, used to compute an equivalent open water advance ratio. This is combined with open-water propeller characteristics, corrected for scale effects, to obtain the thrust estimate. Thrust estimation has been also treated in [10], where the estimate was computed from the propeller torque obtained with a Kalman filter where a linear shaft friction torque was considered. The relation between thrust and torque involved an axial flow velocity model and requires the knowledge of the advance speed. The scheme was also highly sensitive to hydrodynamic and mechanical modeling errors. The performance was validated by simulations. An adaptive observer to estimate shaft speed and thrust was also designed for variable pitch propulsion systems in [3] and [18]. The observer was used for fault detection in the shaft speed control loop. A linear approximation of the propeller characteristics was utilized, therefore this approach could not guarantee accurate results in all four quadrant plane composed by the vessel speed and the propeller shaft speed. Moreover, the observer employed the vessel speed measurement. The contribution of this paper is the development of a fourquadrant thrust estimation scheme, extending the preliminary results described in [22] and [23]. The strength of the presented approach is that only measurements of the propeller shaft speed and the motor torque, normally available on ships, are utilized. Differently from [10], the advance speed which is very difficult to measure in real vessels, is assumed to be unknown.
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Our scheme is based on a robust nonlinear observer to estimate the propeller torque and the shaft speed, and on a mapping to compute the propeller thrust from the observer estimates. The observer is similar to the one introduced in [28], but the inclusion of a nonlinear friction term is a new contribution. Moreover, in order to analyze the effect of the measurement and friction modeling errors on the observer estimates, a Lyapunov based robustness analysis has been performed. The thrust is computed from the torque estimate through a mapping that involves the estimation of the advance ratio. The performance of the proposed scheme is demonstrated by extensive experiments carried out on an electrically driven fixed pitch propeller in a basin with close to open-water conditions, in waves and with a wake screen that scales the local flow down in order to simulate one of the effects of the interaction between the propeller and the vessel hull. The paper is organized as follows. The overall propulsion system is described in Section II. The experimental setup and instrumentation are presented in Section III. Modeling of the propeller shaft dynamics, thrust and torque, and the open-water characteristics is treated in Section IV. The thrust estimation scheme is described in Section V and experimental results are presented in Section VI. Finally, the conclusions are given in Section VII. II. OVERALL S YSTEM D ESCRIPTION
2
III. E XPERIMENTAL S ETUP AND I NSTRUMENTATION The tests were carried out at the Marine Cybernetics Laboratory, an experimental laboratory located at NTNU in Trondheim. The basin, 6.45 m wide, 40 m long and 1.5 m deep, is equipped with a 6DOF towing carriage that can reach a maximum speed of 2 m/s and with a wave generator able to generate waves up to 0.3 m. The tank dimensions may appear too small for accurate open-water and dynamic tests due to the influence of previous motions, presence of walls and free surface motion. The variance of the obtained results was found to be small. We employed a three phase brushless motor in combination with a drive equipped with a built-in torque controller and a build-in shaft speed controller. In this way we could choose to control the motor torque in order to obtain the desired motor torque or the shaft speed to obtain the desired ω. The motor was connected to the propeller shaft through a gear-box with ratio 1:1. The rig with motor, underwater housing, shaft and propeller was attached to the towing carriage in order to move the propeller through the water. The tests were performed on a fixed pitch propeller without duct and with geometric parameters given in Table R was used to interface I. The real-time system Opal RT-Lab R environment to the motor drive and the the Matlab/Simulink sensors. The shaft speed was measured on the motor shaft with a tachometer dynamo. The thrust and torque were measured with an inductive transducer and a strain gauge transducer placed on the propeller shaft, respectively. The measurement of the motor torque was furnished by the motor drive. All the signals were acquired at the frequency of 200 Hz. A sketch of the setup is shown in Fig. 2 and a picture of the propeller system is presented in Fig. 3.
Fig. 1. Block diagram of the propeller system and the thrust estimation scheme.
A block diagram that represents the propeller system and the scheme implemented to estimate the propeller thrust is shown in Fig. 1. The propeller is connected to the motor through a shaft and a gear box. The motor torque applied to the shaft is denoted Qm . The gear ratio is defined by Rgb = ωm /ω, where ωm is the motor shaft angular speed and ω is the propeller angular speed. The value of ω is particularly influenced by the load, represented by the propeller torque Qp , due to the rotation of the blades in the water. The output of the system is the thrust Tp produced by the propeller. The thrust estimation scheme includes a nonlinear observer ˆ p of the propeller load torque that computes the estimate Q and the estimate ω ˆ of the shaft speed. The observer uses the measurements of the motor torque Qm and the propeller shaft speed ω. An estimate Tˆp of the propeller thrust is computed ˆ p and ω using the observer estimates Q ˆ through a mapping f.
Fig. 2.
Sketch of the experimental setup.
TABLE I P1362 PROPELLER GEOMETRICAL PARAMETERS
Parameter D Z P/D Ae /A0
Value 0.25 m 4 1.0 0.55
Description Propeller diameter Number of blades Pitch ratio P/D Expanded blade area ratio
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3
With Rgb = 1 and defining z = Qm − Qp , θ is computed over a time-series of N samples as θ = arg min
N
2
|zi − Qf (ωi ) − Jm ω˙ i | .
(4)
i=1
where the subscript i indicates the i-th sample (see for examples [9]). The parameters obtained are shown in Table II. TABLE II F RICTION MODEL PARAMETERS AND SHAFT MOMENT OF INERTIA .
IV. P ROPELLER M ODELING
A. Propeller shaft dynamics The shaft dynamics is derived by considering the motor connected to the propeller through a rigid shaft and a gearbox with gear ratio Rgb , as shown in the block diagram of Fig. 1. The shaft is considered affected by a friction torque denoted Qf (ω), which is assumed to depend only on the shaft speed. The shaft dynamics can be written as Jm ω˙ = Rgb Qm − Qp − Qf (ω),
(1)
where Jm is total moment of inertia including the shaft, the gear box and the propeller. The friction torque has been modeled as ω
+ kf2 ω + kf3 arctan(kf4 ω), (2) where the Coulomb effect, usually written as a sign(ω), has been replaced by the function π2 arctan( ω ) with a small positive in order to avoid the singularity for ω = 0. The remaining terms in (2) represent a linear and a nonlinear viscous effect. All the coefficients kfi are constant and positive. The static friction model (2) is able to approximate the friction torques experienced in practice (see [1], [16] and [22]). Qf (ω) = kf1 arctan
1) Shaft moment of inertia and friction torque identification: To identify the friction torque and the shaft moment of inertia in (1), we ran tests with different motor torque profiles and various towing carriage speeds. From the measurement of the propeller angular speed, the motor torque and the propeller torque, and computing the derivative of ω with the necessary filtering, we identified the parameters kfi of the friction torque model (2) and the shaft moment of inertia Jm . The parameters kfi and Jm can be grouped in the vector T θ = kf1 kf2 kf3 kf4 Jm . (3)
Value 6.07 · 10−3 3.97 · 10−1 9.28 · 10−3
Parameter kf3 [kg · m2 /s] kf4 [−] [−]
Value 6.61 · 10−3 8.94 · 10−2 1 · 10−3
Figure 4 shows the friction torque computed from measurements and the identified model. The friction exhibits a nonlinear behavior and is affected by the temperature in the gears, bearings and oil. The friction presents also a hysteresis effect but its influence is not very significant and it has been neglected. For the tested propeller system, the losses due to the friction torque are quite high compared to a full scale propeller, where they are usually less significant. 3 2
Qf [N m]
Fig. 3. Propeller open-water configuration (main picture) and wake screen (small picture).
Parameter Jm [kg · m2 ] kf1 [kg · m2 /s] kf2 [kg · m2 /s]
1 0 -1 -2 -3 -80
Measured Qf Qf model -60
-40
-20
0
20
40
60
80
ω [rad/s] Fig. 4. Friction torque: computed from measurements and the identified nonlinear static model.
B. Propeller thrust and torque
Fig. 5. Definition of the advance speed ua and vessel speed U and the undisturbed flow speed uu .
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Modeling the thrust and torque produced by a propeller is a complicated task, since it is difficult to develop a finitedimensional analytical model from the laws of physics. This is mainly due to the difficulty in modelling the flow dynamics, especially when the flow is not uniform [1], [4], [5], [12], [16], [20]. The thrust and torque depend also on the propeller geometric parameters (i.e. propeller diameter, pitch angle, etc.), the non-dimensional parameters advance ratio J and Reynolds number, the propeller submergence and environmental state (waves, currents, etc.). A common practice is the use of simplified models which are chosen based on the propeller application. See for example [8], [21] and [28] and the references therein. 1) Open-Water propeller characteristics: Neglecting the effect of waves and marine currents, and assuming a deeply submerged fixed pitch propeller, the thrust and torque are usually represented in nondimensional form. One form is represented by the standard open-water coefficients KT and KQ , given as functions of the advance ratio J. The term openwater refers to the case where the propeller is tested without the presence of a vessel hull. The coefficients KT and KQ are computed from [6] as 4π 2 , (5) KT = T p ρ |ω| ωD4 KQ = Qp
4π 2 , ρ |ω| ωD5
(6)
where ρ is the density of the water and D is the propeller diameter. The advance ratio is computed as 2πua , (7) ωD where ua is the advance speed, i.e. the water inflow velocity to the propeller. The KT and KQ curves are measured for a range of propeller advance numbers J, usually in a cavitation tunnel or a towing tank [14]. When the propeller is working in water that has been disturbed by the passage of the hull, it is no longer advancing relatively to the water at the speed of the ship U, but at some different speed ua . The advance speed is very difficult to measure and an estimate of ua is usually computed using the steady-state relation J=
ua = (1 − w)U,
(8)
where w is the wake fraction number, often identified from experimental tests (see e.g. [17]). Figure 5 shows a sketch of a vessel with the velocities involved. The surge vessel speed U is relative to the earth while ua is the longitudinal water speed relative to the propeller disc. The undisturbed water velocity uu has the same magnitude as the vessel speed but with opposite direction. A measure of the propeller performance is the open-water efficiency, which is defined as the ratio of the produced to the consumed power by the propeller. The propeller efficiency is usually plotted for positive values of J and is computed from (5), (6) and (7) as η=
u a KT JKT u a Tp = = . ωQp ωDKQ 2πKQ
(9)
4
The curves KT and KQ are usually employed in the first and in the third quadrant of the plane composed by ua and ω, and they are not defined for ω = 0. For propellers operating in the whole plane (ua ,ω), four-quadrant open-water characteristics are normally utilized [6]. The four-quadrant coefficients CT and CQ are plotted as functions of the advance angle β. The value of β is computed with the four quadrant inverse tangent function as β = arctan2 (ua , 0.7Rω) ,
(10)
where R is the propeller disc radius. The four-quadrant coefficients are calculated from [6] as CT = CQ =
Tp , 1 2 2 ρVr A0
(11)
Qp , 1 2 2 ρVr A0 D
(12)
where A0 is the propeller disc area and Vr is the relative advance velocity: Vr2 = u2a + (0.7Rω)2 .
(13)
2) Measured open-water characteristics: To measure the open-water propeller characteristics, we performed tests at different values of the advance ratio J. To obtain the desired shaft speed ω, the built-in speed controller of the drive was used. In our setup, the housing that contains gear and measurement devices does not create a significant wake and the advance speed ua has been considered equal to the towing carriage speed U. This yields a wake fraction number w equal to zero. The standard propeller characteristics are plotted in Fig. 6. In Fig. 7 a sample of filtered data used to derive the KT and KQ curves are plotted. For positive ua and ω, the inflow to the propeller is uniform and the thrust and torque are quite steady. When the advance speed becomes negative, the propeller tries to reverse the inlet flow and a recirculation zone (often called a ring vortex) occurs [30]. This is due to the interaction between the inlet flow and the reversed flow. The flow then becomes unsteady and can cause oscillations in the propeller thrust and torque. For negative values of J, KT and KQ were computed from the average of the measured thrust and torque. From Fig. 6, it can be noticed that the tested propeller is not symmetric in the thrust production with respect to the shaft speed. For positive values of ω the efficiency is higher because the propeller was designed to work mainly at forward vessel speed. Figure 8 shows the four-quadrant propeller characteristics and an approximation computed with a 25th order Fourier series, commonly adopted for the CT and CQ curves [6] . 3) Torque model for the observer: In order to estimate the propeller torque with an observer, a dynamic model for Qp is developed. The propeller torque Qp is treated as a timevarying parameter and modeled as a first order process with a positive time constant τ1 , driven by a bounded noise w1 as in [22], [23] and [26]: 1 Q˙ p = − Qp + w1 . (14) τ1
IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY, VOL. XX, NO. Y, MONTH 2008
5
2.5
1.5
u 0
10KQ KT η
u >0, ω>0
a
a
Fourier CT Fourier −10CQ meas CT meas −10CQ
2
1
1.5 1
CT , −10CQ
0.5
(a) 0 1.5
u >0, ω −kf2 + αl22κ + 2μκ our analysis to values of J in the range [−1.5, 1.1]. However, then the system (19) is input-to-state stable (ISS). when the shaft speed is reversed, the propeller works for a Proof: See Appendix. short time outside the range of J that has been considered. Remark 2: For the observer considered, there always exist In this condition, both thrust and torque are small since the parameter and gain values that can be chosen according to the shaft speed is small, and the approximation error of GQT (J) above criteria. does not affect the estimation significantly. This is shown in The ISS property of the observer error dynamics provides the experimental results reported in Section VI. ˆp the robustness of the observer against noise and modeling The estimation of J is performed using the estimates Q ˆ Q , an estimate of KQ as errors. The observer errors, and thus the torque and shaft and ω ˆ . From (6), we can compute K speed estimates, remain bounded for any initial conditions 2 regardless of the values of the measurement errors, the noise ˆQ = Q ˆ p 4π , ω K ˆ = 0. (27) ρˆ ω 2 D5 w1 in the propeller torque model and the difference between the friction torque model and the actual friction. In particular, The value of K ˆ Q has been limited by the upper bound the observer robustness against friction torque modeling errors K (J ) and lower bound K (J Q min Q max ) where [Jmin , Jmax ] is very important. The shaft friction torque may depend upon is the range of J. In this way we handle also the case when variables which are not directly accounted for in the model, ω ˆ = 0. An estimate Jˆ of the advance ratio can be derived by like temperature and bearing lubrication. ˆ Q computed inverting the KQ curve and using the value of K with (27). It can be noticed in part (a) of Figures 9 and 10, B. Thrust and torque relationship that the KQ curve is not invertible in the whole range of As stated above it is difficult to derive accurate models J considered. For this reason, the J axis has been divided for the thrust and torque, especially when the inflow to in three zones. In zones 1 and 3 the KQ curve is invertible the propeller is not uniform. For example, [1], [11] and and an accurate estimate of J can be computed. In zone 2, [20] experimentally demonstrated the need of including the Jˆ has been approximated with zero to ensure correct thrust
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ˆ for positive and Plots of the gains GQT (J) and GQT (J), ˆp negative values of ω, are shown Figures 9 and 10. If Q ˆ and ω ˆ are accurate, outside zone 2, GQT (J) approximates accurately GQT (J) and the estimated thrust is precise. In ˆ is equal to GQT (0) and, due to the propeller zone 2, GQT (J) ˆ and GQT (J) characteristics, the difference between GQT (J) is not of significant magnitude. In the border of zone 2, ˆ has been joined smoothly to GQT (0) in order to GQT (J) avoid sharp variation on the thrust estimate. For the tested propeller, the value of J is limited to the range [-1.5,1.1] for ω ≥ 0 and to the range [-1.5,0.9] for ω < 0. The maximum ˆ and GQT (J) is about 8% for relative error between GQT (J) ω ≥ 0 and 13% for ω < 0.
J -1.5
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0.25
Zone 1
K Q (ω < 0)
0.2
Zone 2
Zone 3
0.15 0.1 0.05
(a)
0
25
T p /Q p (ω < 0)
estimation when the advance speed is zero, i.e. the vessel is at rest and not subjected to current. This approximation introduces an error on the estimate which is computed as ˆ p GQT (J)| ˆ ωˆ ≥0 Q ω ˆ≥0 Tˆp = (28) ˆ ˆ ωˆ 4.1 · 10−2 + 3.5 · 10−12 l2 , 0 < l2 < 5 · 1020 ,
Remark 3: On full scale vessels, the open-water characteristics obtained in a model scale is expected to be corrected for scale effects [13]. Remark 4: If the propeller open-water characteristics are not available, Computational Fluid Dynamics (CFD) techniques can help to derive it from the 3D drawing of the propeller, see for example [19] and references therein. Remark 5: The thrust and torque relationship is not derived from the four-quadrant propeller characteristics because it is ˆp not possible to estimate the advance angle β only from Q and ω ˆ , since we cannot compute the coefficient CQ without knowing ua . This makes this parameterization difficult to use ˆp. ˆ and Q while we can estimate J from KQ , computed using ω
which practically allows us to choose l1 and l2 freely. Considering the fact that we want the observer dynamics faster than the system dynamics and at the same time, too high gains can produce oscillatory estimates due to the measurement noise on the shaft speed, l1 and l2 were chosen as a trade-off between the two opposite requirements. The time constant was obtained from a sensitivity analysis on the observer estimation errors with respect to τ1 . Running the observer with l1 = 3 kg · m2 /s, l2 = 80 kg · m2 /s2 , on data acquired over more than 1 h of tests carried out in open-water conditions at different advance speeds and shaft speeds, we derived Fig. 12. The graph shows the root mean square error (RMSE) between the observer estimates and the
IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY, VOL. XX, NO. Y, MONTH 2008
2
ω ˜ ˜p Q
RSME
1.5
60
ω [rad/s]
20 0 -20 -40
(a)
-60 4 2 0 -2 -4
(b)
6 measured Qp ˆp Q
1
4
0
0.01
0.1
1
10
Q p [N m]
0.5
100
τ1 [s]
B. Open water tests A series of tests to validate the proposed scheme have been carried out in open-water conditions. The propeller was submerged at h/R = 4, where h is the propeller submergence, in order to avoid losses due to ventilation. Figures 13 and 14 show data from an experiment where both the advance speed and the shaft speed had a trapezoidal form. Figures 15 and 16 show data from a test with sinusoidal advance and shaft speed. The shaft speed and propeller torque estimates are plotted in part (a) and (c) of Figures 13 and 15, respectively. Both estimates are accurate and almost indistinguishable from the measurements. The observer estimation errors are plotted in part (b) and (d) of Figures 13 and 15. The advance speed is plotted in part (e) of the same figures. Figures 14 and 16 show the measured and estimated thrust from the same tests. The estimated Tˆp , shown in part (a), obtained with the proposed method reproduces quite well the measurements in all the quadrants. Part (e) of the same figures shows the advance angle β. The estimate Tˆp is compared with the estimate TˆpCT , shown in part (c), computed using the measured four-quadrant propeller characteristics CT , introduced in Section IV, with ua = U. This estimate is not very accurate, especially in the 2nd and 4th quadrant where the inflow to the propeller can be irregular. The proposed scheme furnish a more accurate thrust estimate since it can sense the effect of the flow variation through the propeller torque estimate. The thrust estimation errors are shown in (b) and (d) of Figures 14 and 16. C. Wake screen test Tests with a wake screen, shown in Fig. 3, were performed to simulate one of the effects of the hull on the propeller inflow. The wake screen created an uniform loss of speed. This does not represent entirely the effect of the hull because
ˆ p [Nm] Qp − Q
Observer estimation errors for different values of τ1 .
2 0 -2 -4
(c)
-6 1 0.5 0 -0.5 -1
(d)
2
u a [m/s]
Fig. 12.
measured ω ω ˆ
40
ω−ω ˆ [rad/s]
measurements. The value of the time constant has been varied between 0.01 and 100. For τ1 ≥ 1, the accuracy of the estimates is practically the same, while for smaller values the estimate are less precise. For small values of τ1 , the torque estimation error decreases when the shaft speed estimation error diminishes. This allows us to choose the time constant based on the speed error, since the torque measurements is not available in real cases. The observer parameters used in the experiments are l1 = 3 kg · m2 /s, l2 = 80 kg · m2 /s2 and τ1 = 10 s.
8
(e)
1 0 -1 -2
795
805
815
825
835
Time [s] Fig. 13.
Data from the first open-water test.
the propeller inflow is not usually uniform. The propeller was submerged at h/R = 4. Figure 17 shows the results from a test where both the advance speed and the shaft speed had a trapezoidal form. As for the open-water experiments, the estimates provided by the observer, shown in (a) and (b), are very accurate. In (d) and (e) of the same plot, the estimate Tˆp obtained with the proposed method is compared with the estimate TˆpCT , computed using the measured four-quadrant propeller characteristic CT . The advance speed is computed with (8), where the value of the wake fraction w has been identified from tests performed in steady-state conditions. For positive towing carriage speed U , shown in (c), the experimentally found value was w = 0.3. For negative towing carriage speed, the inlet water flow to the propeller was not affected by the grid, placed upstream of the propeller, and the wake fraction number was zero. The estimate Tˆp is quite accurate also in this experiment while the estimate TˆpCT , as for the open-water tests, is accurate only
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40 0 -40
0 -20
(a)
-60
ω−ω ˆ [rad/s]
Tp − Tˆp [N]
-120 60 30 0 -30 -60
20
-40
(a)
-80
measured ω ω ˆ
40
ω [rad/s]
80
T p [N ]
60
measured Tp Tˆp
120
9
(b) measured Tp Tˆp
120
4 2 0 -2 -4 6
(b) measured Qp ˆp Q
4
CT
Q p [N m]
T p [N ]
80 40 0 -40
ˆ p [Nm] Qp − Q
Tp − TˆpCT [N]
-120 60 30 0 -30 -60 360
(d)
β [deg]
rd
3
180 nd
2
(e)
st
1
0
795
805
815
-2
(c)
-6 1 0.5 0 -0.5 -1 2
u a [m/s]
th
4
270
90
0
-4
(c)
-80
2
835
(e)
1 0 -1 -2 1200
825
(d)
1210
1220
1230
1240
1250
1260
1270
1280
Time [s]
Time [s] Fig. 15. Fig. 14.
Data from the second open-water test.
Data from the first open-water test.
when ω and the ua have the same sign. D. Test in waves Tests in waves were carried-out in order to validate the estimation scheme with a periodic propeller inflow and with large losses due to ventilation. Figure 18 shows the result of a test performed in regular waves with amplitude 0.05 m (equivalent to 0.2 D) and frequency 0.69 Hz. The propeller was also moved along its vertical axis with a sinusoidal motion. This was done to simulate the motion that a propeller may experience in rough sea conditions. This test does not reproduce entirely rough sea conditions but it may be a valid indication of the performance of the proposed method when operating in off-design conditions. The towing carriage was kept at rest (U = 0), but the advance speed was still not zero since the waves create an inflow to the propeller. Part (c) of Fig. 18 shows the propeller vertical displacement d. The propeller shaft speed, depicted in part (a), has been kept constant at 38 rad/s. A drop of thrust and torque occurred when the propeller rotated close to the water surface, since
the load decreased due to ventilation. The small oscillations of thrust and torque were due to waves that created a periodic additional axial velocity component that varied with depth and time across the propeller plane. Both phenomena were well reproduced by the torque estimate, as shown in part (b) of Fig. 18. The thrust estimate Tˆp obtained from the proposed method is depicted in Fig. 18 (d) and is compared with TˆpCT , the thrust computed directly from the CT characteristic. The proposed method produced a satisfactory estimate and both the oscillations and the drop of thrust were properly captured. The estimate TˆpCT has been computed assuming ua = 0, the best guess we could make since U = 0. Since ω was constant and ua = 0, the estimate TˆpCT was constant and could not capture the thrust variations due to waves. VII. C ONCLUSION A thrust estimation scheme for a marine propeller has been developed and experimentally tested on an electrically driven propeller. Tests were performed in open-water conditions, with a wake screen to simulate one of the effects of the hull on the propeller inflow and in waves with vertical propeller motion in
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ω [rad/s]
measured Tp Tˆp
120
T p [N ]
80 40 0
(a)
-80
Tp − Tˆp [N]
-120 60 30 0 -30 -60
U [m/s]
(b)
measured Tp TˆpCT
120
T p [N ]
80
Tp [N ]
40 0
-120 60 30 0 -30 -60
Tp [N ]
(c)
-80
Tp − TˆpCT [N]
measured ω ω ˆ
(a)
(b)
(d)
360 4th
β [deg]
270 rd
3
180 2nd
90
(e)
st
1
0 1200
1210
1220
1230
1240
0 -1
(c)
120 80 40 0 -40 -80 -120 120 80 40 0 -40 -80 -120 360 270 180 90 0 161
1250
1260
1270
1280
measured Qp ˆp Q
1
-2
-40
β [deg]
60 40 20 0 -20 -40 -60 6 4 2 0 -2 -4 -6 2
Qp [N m]
-40
10
(d)
measured Tp Tˆp
(e)
measured Tp TˆpCT
4th 3rd 2nd
(f)
1st
171
181
Time [s]
Time [s] Fig. 16.
Data from the second open-water test.
order to reproduce the motion that a propeller may experience in rough sea conditions. The robustness of the scheme with respect to modeling and measurement errors was demonstrated with the use of Lyapunov theory and corroborated by experimental results. The scheme involved a nonlinear observer to estimate the propeller torque and the shaft speed using only the measurements of the motor torque and the propeller shaft speed. The thrust estimate was computed from the estimated propeller torque and shaft speed involving the estimation of the advance ratio J. This only required knowledge of the standard propeller characteristics. The thrust obtained from the proposed method was compared with the thrust computed from the four-quadrant propeller characteristics showing improved accuracy in the estimates. The thrust estimation scheme can be implemented also for ducted propellers, where the standard propeller characteristics are slightly different compared to a propeller without a duct. Although the presented results concern tests carried out on an electrically driven propeller, the scheme could be applied
Fig. 17.
Data from the wake screen experiment.
also to propellers driven by diesel motors where the motor torque can be measured with strain gauges on the motor shaft [2] or by measuring the fuel index [3]. The presented results are promising for the use of such a thrust estimation scheme in high performance propeller controllers. VIII. ACKNOWLEDGMENTS The authors acknowledge with gratitude Professor Thor Inge Fossen for valuable suggestions and discussions. Matthias Schellhase is gratefully acknowledged for the help given during the experimental tests. The Research Council of Norway is acknowledged as the main sponsor of this project. A PPENDIX The appendix presents the proof of Proposition 1. First we consider the input u, defined in (18), equal to zero ∀t and later we investigate its effect on the error dynamics. Taking the Lyapunov function candidate V := 12 e˜T P1 e˜, where P1 = P1T > 0 and
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(a)
ω [rad/s]
60
V˙ ≤
measured ω ω ˆ
40
−p12 g1 (˜ e1 , ω ˆ )˜ e2 .
Qp [Nm]
4
(33)
on the last term of (33) obtaining (b)
2
measured Qp ˆp Q
−p12 g1 (˜ e1 , ω ˆ )˜ e2
80
(d) 40 measured Tp Tˆp
(e) 40 measured Tp TˆpCT
Data from the experiment with waves.
P1 =
p11 p12
p12 p22
With this choice of l1 , we observe from (36) that p22 > 0 and the inequality (38) certainly holds if 2 μ< . (40) Jm
,
(29)
we can compute its time derivative along the trajectory of (19) obtaining
p12 V˙ = − J1m p11 (kf2 + l1 ) − p12 l2 e˜21 − pτ22 e˜22 + J m
1 − pJ12 (kf2 + l1 ) + pτ12 + pJ11 − p22 l2 e˜1 e˜2 m 1 m −p12 ψ(˜ e1 , ω ˆ )˜ e2 − p11 ψ(˜ e1 , ω ˆ )˜ e1 . (30) ˆ ) we can subtract the linear From the nonlinearity ψ(˜ e1 , ω function α˜ e1 , where α is constant and satisfies A1, such that g1 (˜ e1 , ω ˆ ) = ψ(˜ e1 , ω ˆ ) − α˜ e1 .
(31)
Since the graph of ψ(˜ e1 , ω ˆ ) belongs to the sector [0, α], the ˆ ) belongs to the sector [−α, α], i.e. graph of g(˜ e1 , ω 2
ω : [g(˜ e1 , ω ˆ )] < ∀˜ e1 , ∀ˆ
(34)
Choosing p11 = p12 α2 κJm with κ > 0, the inequality (37) is satisfied for 1 l2 . (39) l1 > −kf2 + 2 + α κ 2μκ
20
Time [s]
2 p12 e1 , ω ˆ )] 2μ [g1 (˜ p12 2 2 ˜1 . 2μ α e
the cross-term in (35) is cancelled. To obtain a negative definite V˙ the following are needed: 1 p12 2 α > 0, p11 (kf2 + l1 ) − p12 l2 − (37) Jm 2μ 1 p22 μ + p12 − > 0. (38) τ1 Jm 2
80
0 10
≤ μ p212 e˜22 + ≤ μ p212 e˜22 +
Using (34) in (33) we attain
12 V˙ ≤ − J1m p11 (kf2 + l1 ) − p12 l2 − p2μ α2 e˜21
p12 p11 − pJ12 (k + l ) + + − p l + p α e˜1 e˜2 f 1 22 2 12 2 τ J m
1 m − pτ22 + pJ12 − μ p212 e˜22 . 1 m (35) Selecting l2 > 0 and p22 such that
1 p12 p12 p11 (kf2 + l1 ) + + + p12 α , (36) p22 = l2 Jm τ1 Jm
(c)
0
Tp [N]
Choosing p12 > 0, we can use Young’s inequality 1 −2xy ≤ μx2 + y 2 ∀μ > 0, μ
0 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Tp [N]
d [m]
1 e˜22 p11 (kf2 + l1 ) − p12 l2 e˜21 − pτ22 + pJ12 1 m Jm − pJ12 (kf2 + l1 ) + pτ12 + pJ11 − p22 l2 + p12 α e˜1 e˜2 m 1 m
−
20 0
Fig. 18.
11
α2 e˜21 .
(32)
Substituting (31) in (30) and recalling that, since p11 > 0, p11 ψ(˜ e1 , ω ˆ )˜ e1 > 0 we get
Combining (39) and (36), we get p22 > l2pJ11m . This yields
p11 p11 2 κJ α m P1 > . (41) p11 p11 α2 κJm
l2 Jm
If l2 < α κ Jm then P1 is positive definite. Choosing the observer gains according to A4 and A5 of Proposition 1, the derivative of the Lyapunov function candidate is negative definite since 2 (42) e , V˙ ≤ − min{q1 , q2 } ˜ 4 2
2
where
1 p12 2 α , p11 (kf2 + l1 ) − p12 l2 − Jm 2μ 1 p22 μ q2 = + p12 − . τ1 Jm 2
q1 =
(43) (44)
The observer error dynamics, with u = 0 ∀t, is thus globally exponentially stable (GES). When the input u is different from zero for some t, the term 2˜ eT P1 B1 u must be added to the
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derivative of the Lyapunov function in (42): 2
≤ − min{q1 , q2 } ˜ e2 + 2˜ eT P1 B1 u 2 ≤ − min{q1 , q2 } ˜ e2 + 2 e˜T P1 B1 u2 2 ≤ − min{q1 , q2 } ˜ e2 + 2 e˜T 2 P1 2 B1 2 u2 . (45) With 0 < θ < 1, we obtain V˙
2
e2 + P1 2 B1 2 ˜ e2 u2 ≤ − min{q1 , q2 } ˜ 2 2 ≤ −(1 − θ) min{q1 , q2 } ˜ e2 − θ min{q1 , q2 } ˜ e2 + P1 2 B1 2 ˜ e2 u2 . (46) For any ˜ e2 such that V˙
˜ e2 ≥ ρ(u2 ), where ρ(u2 ) =
P1 2 B1 2 u2 θ min{q1 , q2 }
(47) (48)
is a (linear) class K function, we obtain 2 V˙ ≤ −(1 − θ) min{q1 , q2 } ˜ e2 ≤ 0.
(49)
Since V is positive definite and radially unbounded, from Theorem 4.19 in [15], the system (19) is ISS. Furthermore, the observer erroris uniformly ultimately bounded (UUB) by ρ supt>t0 (u2 ) . R EFERENCES [1] R. Bachmayer, L. L. Whitcomb, and M. A. Grosenbaugh, “An accurate four-quadrant nonlinear dynamical model for marine thrusters: Theory and experimental validation,” IEEE Journal of Oceanic Engineering, vol. 25, no. 1, pp. 146–159, January 2000. [2] M. Blanke, “Ship propulsion losses related to automatic steering and prime mover control.” Ph.D. dissertation, Technical University of Denmark, 1981. [3] M. Blanke, R. Izadi-Zamanabadi, and T. F. Lootsma, “Fault monitoring and reconfigurable control for ships propulsion plant,” Journal of Adaptive Control and Signal Processing, vol. 12, pp. 671–688, 1998. [4] M. Blanke, K. Lindegaard, and T. I. Fossen, “Dynamic model for thrust generation of marine propellers,” in 5th IFAC Conference of Manoeuvring and Control of Marine craft (MCMC), Aalborg, Denmark, 2000, pp. 363–368. [5] J. P. Breslin and P. Andersen, Hydrodynamics of Ship Propellers. Cambridge University Press, 1994. [6] J. S. Carlton, Marine Propellers and Propulsion. Oxford, U.K.: Butterworth-Heinemann Ltd., 1994. [7] T. I. Fossen, Marine Control Systems - Guidance, Navigation, and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, Trondheim, Norway, 2002. [8] T. I. Fossen and M. Blanke, “Nonlinear output feedback control of underwater vehicle propellers using feedback from estimated axial flow velocity,” IEEE Journal of Oceanic Engineering, vol. 25, no. 2, pp. 241–255, April 2000. [9] P. Gill, W. Murray, and M. Wright, Practical Optimization. London, Academic Press, 1981. [10] C. Guibert, E. Foulon, N. A¨ıt-Ahmed, and L. Loron, “Thrust control of electric marine thrusters,” in 31nd Annual Conference of IEEE Industrial Electronics Society. IECON 2005, Raleigh, North Carolina, USA, 6-10 November 2005. [11] A. J. Healey, S. M. Rock, S. Cody, D. Miles, and J. P. Brown, “Toward an improved understanding of thruster dynamics for underwater vehicles,” Symp. Autonomous Underwater Vehicle Technology, Boston MA, pp. 340–352, 1994. [12] ——, “Toward an improved understanding of thruster dynamics for underwater vehicles,” IEEE Journal of Oceanic Engineering, vol. 20, no. 4, pp. 354–61, October 1995. [13] ITTC, Recommended Procedures - Performance, Propulsion 1978 ITTC Performance Prediction Method - 7.5-02-03-01.4, International Towing Tank Conference, 1999.
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